# On the size of $k$-cross-free families

## TL;DR

This paper improves the upper bound on the size of families of subsets with no $k$ pairwise crossing sets from $O_k(n \, \log n)$ to $O_k(n \log^{*} n)$, refining a longstanding conjecture in combinatorics.

## Contribution

The authors establish a tighter upper bound of $O_k(n \log^{*} n)$ for the size of $k$-cross-free families, advancing previous bounds and addressing a 40-year-old conjecture.

## Key findings

- Bound improved from $O_k(n \log n)$ to $O_k(n \log^{*} n)$
- Confirms the conjecture for larger $k$ with a nearly optimal bound
- Uses novel combinatorial techniques to refine existing bounds

## Abstract

Two subsets $A,B$ of an $n$-element ground set $X$ are said to be \emph{crossing}, if none of the four sets $A\cap B$, $A\setminus B$, $B\setminus A$ and $X\setminus(A\cup B)$ are empty. It was conjectured by Karzanov and Lomonosov forty years ago that if a family $\mathcal{F}$ of subsets of $X$ does not contain $k$ pairwise crossing elements, then $|\mathcal{F}|=O_{k}(n)$. For $k=2$ and $3$, the conjecture is true, but for larger values of $k$ the best known upper bound, due to Lomonosov, is $|\mathcal{F}|=O_{k}(n\log n)$. In this paper, we improve this bound by showing that $|\mathcal{F}|=O_{k}(n\log^{*} n)$ holds, where $\log^{*}$ denotes the iterated logarithm function.